Clausius relation and Friedmann equation in FRW universe model

TL;DR

This paper explores the thermodynamic interpretation of the Friedmann equation in FRW universe models, emphasizing the role of Kodama observers and extending the analysis to Hořava-Lifshitz gravity variants.

Contribution

It demonstrates that the Clausius relation and Friedmann equations can be derived from thermodynamics using Kodama vectors, extending previous work to new gravity theories.

Findings

Energy flux through the apparent horizon is determined by the Kodama vector.

The thermodynamic derivation applies to Hořava-Lifshitz gravity and its IR modifications.

The Clausius relation is relative to Kodama observers in FRW universe.

Abstract

It has been shown that Friedmann equation of FRW universe can be derived from the first law of thermodynamics in Einstein gravity, Gauss-Bonnet gravity, Lovelock gravity, scalar-tensor gravity and $f(R)$ gravity. Moreover, it was pointed out that the temperature of the apparent horizon can be obtained using the tunneling formalism for the corresponding observers defined by Kodama vector. In this article, we find that the energy flux through the apparent horizon can be determined by using the Kodama vector. This implies the fact that the Clausius relation and the first law of thermodynamics associated with the apparent horizon in FRW universe is relative to the Kodama observers.We illustrate the derivation of Friedmann equation, and also extend the study to the cases of Ho\v{r}ava-Lifshitz gravity and IR modified Ho\v{r}ava-Lifshitz gravity.